\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 142, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/142\hfil Coupled system of Maxwell and heat equations]
{Orthogonal decomposition and asymptotic behavior 
for a linear coupled system of\\  Maxwell and heat equations}

\author[C. Buriol, M. V. Ferreira \hfil EJDE-2015/142\hfilneg]
{Celene Buriol, Marcio V. Ferreira}

\address{Celene Buriol \newline
Department of Mathematics,
Federal University of Santa Maria,
Santa Maria, CEP 97105-900, RS, Brazil}
\email{celene@smail.ufsm.br}

\address{Marcio V. Ferreira \newline
Department of Mathematics,
Federal University of Santa Maria,
Santa Maria, CEP 97105-900, RS, Brazil}
\email{marcio.ferreira@ufsm.br}

\thanks{Submitted June 30, 2014. Published May 21, 2015.}
\subjclass[2010]{35Q61, 35Q79, 35B40}
\keywords{Maxwell equation; orthogonal decomposition; exponential decay}

\begin{abstract}
 We study the asymptotic behavior in time of the solutions of a coupled
 system of linear Maxwell equations with thermal effects.
 We have two basic results. First, we prove the existence of a strong
 solution and obtain the orthogonal decomposition of the electromagnetic field.
 Also, choosing a suitable multiplier, we show that the total energy of
 the system decays exponentially as $t \to + \infty$. The results obtained
 for this linear problem can serve as a first attempt to study other nonlinear
 problems related to this subject.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks


\section{Introduction}

It is undisputed the growing interest in understanding phenomena involving 
processes of reciprocal action between variations in the electromagnetic 
field in a region and the temperature or even other situations that are 
related to electromagnetic waves propagation (see \cite{2,4,11,14}).

In this work we consider a coupled system that describes interactions of 
the electromagnetic field with the temperature variation governed
by the linear model
\begin{gather}
\epsilon \mathbf{E}_{t} - \nabla \times \mathbf{H} + \sigma(x) \mathbf{E}
+ \gamma \nabla \theta  =0\quad\text{in }\Omega \times (0,+\infty),\label{eq1.1}\\
\mu \mathbf{H}_{t} + \nabla \times \mathbf{E}  =0\quad
\text{in }\Omega \times (0,+\infty),
 \label{eq1.2}\\
\theta_{t} -\operatorname{div}( \nabla \theta - \lambda \mathbf{E} ) =0
\quad\text{in }\Omega \times (0,+\infty),\label{eq1.3}\\
\operatorname{div}(\mu \mathbf{H} ) =0\quad\text{in }\Omega 
\times (0,+\infty)\label{eq1.4}
\end{gather}
with initial and boundary conditions
\begin{gather}
\mathbf{E}(x,0)=\mathbf{E}_0(x),\quad \mathbf{H}(x,0)=\mathbf{H}_0(x)\quad 
\text{and}\quad
\theta(x,0)=\theta_0(x)\quad \text{in }\Omega,\label{eq1.5}
\\
\eta \times \mathbf{E}=0,\quad  \eta \cdot \mathbf{H}=0,\quad \theta=0 \quad
\text{on } \Gamma \times(0,+\infty).
\label{eq1.6}
\end{gather}
Here $\Omega$ is a bounded, open, simply-connected domain of $\mathbb{R}^3$ 
with a regular boundary $\Gamma = \partial \Omega$. 
The functions $\mathbf{E}=\mathbf{E}(x,t)=\left( E_1(x,t),E_2(x,t),E_3(x,t)\right)$,
$\mathbf{H}=\mathbf{H}(x,t)=\left( H_1(x,t),H_2(x,t),H_3(x,t)\right)$ and
$\theta = \theta(x,t)$ (hereafter, a bold letter means a vector or a vector 
function in $\mathbb{R}^3$) represent, respectively, the electric field, 
the magnetic field and the difference of temperature between the
actual state and a reference temperature at location $x\in \Omega$ 
and time $t$. In \eqref{eq1.6}, $\eta$ is the outward normal on $\Gamma$.
In \eqref{eq1.1}-\eqref{eq1.2}, $\nabla \times \mathbf{v}$  indicates
the curl of the vectorial function $\mathbf{v}$ and $\epsilon$ and $\mu$
 are positive constants characteristics of the medium considered called, 
respectively, the permittivity and
the magnetic permeability. $\sigma=\sigma (x)$ is a real valued 
$L^\infty(\Omega)$-function representing the electric conductivity (see \cite{7}),
related with the Ohm's law and satisfies the hypothesis
\begin{equation}\label{eq1.7}
\sigma_0 \le  \sigma(x) \le \sigma_1,
\end{equation}
where $\sigma_0$  and $\sigma_1$ are positive constants. 
Moreover, $\gamma$ and $\lambda$ are coupling constants which, 
for simplicity, we will assume positive.

The mathematical model \eqref{eq1.1}-\eqref{eq1.4} is motivated by 
considering the classical Maxwell's equations that are coupled to
a heat equation, modeling an expectedly interaction of the electromagnetic 
field with the temperature variation in the bounded domain $\Omega$ 
with perfectly conducting  boundary  $\Gamma = \partial \Omega$. 
In fact, if $\mathbf{E}(x,t)$ and $\mathbf{H}(x,t)$ denote the electric and
magnetic fields in $\Omega$, respectively, and $\mathbf{D}(x, t)$ and
$\mathbf{B}(x, t)$ are the electric displacement and magnetic induction in
$\Omega$, respectively, then hold (see \cite{7}) the \emph{Faraday's law}
\begin{equation}\label{eq1.01}
\nabla \times \mathbf{E}  =-\mathbf{B}_{t},
\end{equation}
the \emph{Ampere's law}
\begin{equation}\label{eq1.02}
\nabla \times \mathbf{H}  =\mathbf{J}+\mathbf{D}_{t},
\end{equation}
where $\mathbf{J}$ represents the current density, and the 
\emph{Gauss's law for magnetism}
\begin{equation}\label{eq1.03}
\operatorname{div}\mathbf{B}=0.
\end{equation}
In our case, we  assume the constitutive relations
\begin{equation}\label{eq1.04}
D= \epsilon \mathbf{E},\quad B=\mu \mathbf{H}
\end{equation}
and \emph{Omh's law}
\begin{equation}\label{eq1.04b}
\mathbf{J}= \sigma \mathbf{E}
\end{equation}
and take, for simplicity, $\epsilon$ and $\mu$ positive constants. 
The boundary condition \eqref{eq1.6} is consistent with the fact that 
the boundary $\Gamma$ is perfectly conducting, such that the tangential 
component of the electric field must vanish.

The model for the propagation of heat turns into well-known equations 
for the temperature $\theta$ (difference to a fixed constant reference 
temperature) and the
heat flux vector $\mathbf{q}$,
\begin{equation} \label{eq1.05}
\theta_{t} + \rho\operatorname{div}\mathbf{q} = 0
\end{equation}
and
\begin{equation} \label{eq1.06}
\mathbf{q} + \kappa\nabla\theta = 0,
\end{equation}
where $\rho$ and $\kappa$ are positive constants. Equation \eqref{eq1.05}
represents the assumed Fourier law of heat conduction.
Replacing \eqref{eq1.06} into \eqref{eq1.05} we obtain the parabolic heat equation
\begin{equation} \label{eq1.07}
\theta_{t} - \rho\kappa\Delta\theta = 0.
\end{equation}
The system we consider is composed by the Maxwell's equations
\eqref{eq1.01}-\eqref{eq1.04} that are coupled to
a heat equation \eqref{eq1.07}
modeling an expectedly interaction effect through heat conduction.
Indeed, we consider the problem \eqref{eq1.1}-\eqref{eq1.6}.
We point out that the results obtained for this linear problem can serve
as a first attempt to study, for example, the stabilization of solutions
of the nonlinear problems of inductive heating or microwave heating
(see \cite{14,15}).

It is worth note that inductive and microwave heating processes are gaining 
increasing acceptance in industry (metal hardening and preheating for forging 
operations, for example) and in some fields of science, such as biomedical
 engineering (see \cite{6,8,9}). Some of these processes are modeled mathematically 
by nonlinear systems of Maxwell's equations coupled with the heat equation 
(see \cite{8,9,15}). Such systems have been studied by some authors not only with
 respect to the existence of solution, like \cite{14,15}, but also with respect 
to the regularity of the solution and blow up properties.
To the best of the authors knowledge, little is known about the asymptotic 
behavior of the energy associated with such nonlinear models. The cases analyzed, 
in general, are limited to those in one dimension (see \cite{5,10}). 
Hence the importance of studying the behavior of the solution of a 
mathematical model, even in the linear case, involving Maxwell's equations 
under thermal effects, which is presented in this paper.

Concerning the system \eqref{eq1.1}-\eqref{eq1.6}, the 
\emph{Total Energy} is given by
$$
\mathcal{E} (t)= \frac{1}{2} \int_{\Omega}(\lambda \epsilon |\mathbf{E}|^2
 + \lambda \mu |\mathbf{H}|^2 + \gamma|\theta|^2)\,dx,
$$
where $| \mathbf{E}| ^{2}=\sum_{j=1}^{3}E_{j}^{2}$ and
$| \mathbf{H}| ^{2}=\sum_{j=1}^{3}H_{j}^{2}$.
Formally, an easy calculation gives us that the derivative of $\mathcal{E} (t)$ 
is given by
$$
\frac{d \mathcal{E} (t)}{dt} =  - \lambda  \int_{\Omega}\sigma(x)|\mathbf{E}|^2 \,dx
- \gamma  \int_{\Omega}|\nabla \theta|^2 \,dx\leq 0.
$$
Therefore one may ask, ``Does $\mathcal{E} (t)\to 0$ as $t\to +\infty$?'', 
and if this is the case,
``Does $\mathcal{E} (t)\to 0$ decay at a uniform rate as $t\to +\infty$?'' 
This is not difficult to answer
in the case of Maxwell's equations with the dissipation given by the conductivity 
$\sigma$ with hypothesis \eqref{eq1.7}. In fact, this case lead to dissipative
 wave equations for the electric field $\mathbf{E}$ and
 the magnetic field $\mathbf{H}$, which have exponential decay.
In our case the uniform stabilization of system \eqref{eq1.1}-\eqref{eq1.6} 
requires a more detailed discussion, which we present in this article.

This article is organized as follow. In section 2 we present some functional 
spaces and basic results. In section 3 we obtain the strong global solution 
of system \eqref{eq1.1}-\eqref{eq1.6}. To obtain the exponential decay 
of the energy, in section 4 we obtain a special decomposition of the 
electromagnetic field in suitable Sobolev spaces. 
Finally, section 5 is devoted to study the exponential decay of the total 
energy associated to system \eqref{eq1.1}-\eqref{eq1.6}.


\section{Basic definitions and preliminary results}
\label{sec:1}

In this section we introduce some standard functional spaces as defined in 
\cite{1,2,3}.
Hereafter the bracket $(\cdot , \cdot)$ and $\| \cdot \|$ will denote,
 respectively, the standard inner product and norm of $L^2(\Omega)^3$
 or $L^2(\Omega)$.
Let
\begin{gather*}
 H(\operatorname{curl},\Omega) = \{ \mathbf{v} \in L^2(\Omega)^3 ;
 \nabla \times \mathbf{v} \in  L^2(\Omega)^3 \},
\\
H(\operatorname{div},\Omega) = \{ \mathbf{v} \in L^2(\Omega)^3 ;
  \operatorname{div} \mathbf{v} \in  L^2(\Omega) \},
\end{gather*}
Hilbert spaces with their respective inner products
\begin{gather*}
\left(\mathbf{u},\mathbf{v} \right)_{H(\operatorname{curl},\Omega)}
=\left(\nabla \times\mathbf{u},\nabla \times\mathbf{v} \right)
+\left(\mathbf{u},\mathbf{v} \right),
\\
\left(\mathbf{u},\mathbf{v} \right)_{H(\operatorname{div},\Omega)}
=\left(\operatorname{div}\mathbf{u},div\,\mathbf{v} \right)
+\left(\mathbf{u},\mathbf{v} \right).
\end{gather*}
Let $H_0(\operatorname{curl},\Omega)$ be the closure of
$$
  \{ \mathbf{v} \in H(\operatorname{curl},\Omega)
\cap C^{1}(\overline{\Omega});   \mathbf{v} \times \eta=0 \text{ on }  \Gamma \}
$$
in $H(\operatorname{curl},\Omega)$ and let $H_0(\operatorname{div},\Omega)$
be the closure of
$$
  \{ \mathbf{v} \in H(\operatorname{div},\Omega)\cap C^{1}(\overline{\Omega}) ;
  \mathbf{v} \cdot \eta=0\text{ on } \Gamma \}
$$
in $H(\operatorname{div},\Omega)$.

To obtain the result of existence of solution we still need to define the 
following spaces
\[
 H(\operatorname{div}0,\Omega) = \{ \mathbf{v} \in L^2(\Omega)^3 ;
\operatorname{div} \mathbf{v} =0\}
\]
and the closed subspace of the Hilbert space $L^2(\Omega)^3$
\[
H_0(\operatorname{div}0,\Omega) = \{ \mathbf{v} \in H(\operatorname{div}0,\Omega) ;
 \mathbf{v}\cdot \eta  =0 \text{ on }\Gamma \} 
= H_0(\operatorname{div},\Omega) \cap H(\operatorname{div}0,\Omega).
\]


\begin{lemma} \label{lema2.1}
Let  
$P_0:L^2(\Omega)^3 \to  H_0(\operatorname{div}0,\Omega)$ be
the projection operator defined by 
$$ 
\mathbf{u} \to P_0\mathbf{u}=\mathbf{u}_1,
$$
where $\mathbf{u}=\mathbf{u}_1+\mathbf{u}_2$,
with  $\mathbf{u}_1 \in H_0(\operatorname{div}0,\Omega)$ and 
$\mathbf{u}_2 \in H_0(\operatorname{div}0,\Omega)^{\bot}$.
We have the following statements:
\begin{itemize}
\item[(i)]  $ P_0(H(\operatorname{curl},\Omega)) \subset 
H(\operatorname{curl},\Omega) \cap H_0(\operatorname{div}0,\Omega)$;

\item[(ii)] $ H(\operatorname{curl},\Omega)\cap H_0(\operatorname{div}0,\Omega) $ 
is dense in $H_0(\operatorname{div}0,\Omega)$.
\end{itemize}
\end{lemma}

\begin{proof} To prove  (i) it is sufficient to show that 
$P_0(H(\operatorname{curl},\Omega)) \subset H(\operatorname{curl},\Omega)$. 
To this we use a similar idea as in \cite{11}. 
Let $\mathbf{u} \in H(\operatorname{curl},\Omega)$. Setting  
$\Psi \in {\mathcal{D}}(\Omega)^3$, we have
\begin{align*}
\langle \nabla \times (P_0\mathbf{u}), \Psi \rangle  
&=  \langle P_0\mathbf{u}, \nabla \times\Psi \rangle  \\
& =  \int_{\Omega}P_0\mathbf{u} \cdot \nabla \times \Psi\,dx 
= \int_{\Omega}\mathbf{u} \cdot \nabla \times \Psi\,dx  \\
& =  \langle \mathbf{u} , \nabla \times \Psi \rangle
 = \langle \nabla \times \mathbf{u} ,  \Psi \rangle,
\end{align*}
for all  $ \Psi \in {\mathcal{D}}(\Omega)^3$, where we have used that 
$\nabla \times \Psi \in H_0(\operatorname{div}0,\Omega)$.

The previous identity give us 
$\nabla \times(P_0\mathbf{u}) = \nabla \times \mathbf{u} \in L^2(\Omega)^3$. 
This proves (i).

(ii) By (i) we have
\[
P_0( {\mathcal{D}}(\Omega)^3 ) \subset H(\operatorname{curl},\Omega) 
\cap H_0(\operatorname{div}0,\Omega) \subset H_0(\operatorname{div}0,\Omega),
\]
so to prove (ii) it is sufficient to prove that $P_0( {\mathcal{D}}(\Omega)^3 ) $ 
is dense in  $H_0(\operatorname{div}0,\Omega)$.

Let $\mathbf{v} \in H_0(\operatorname{div}0,\Omega)$. 
So $\mathbf{v} \in L^2(\Omega)^3$, and there exist a sequence 
$(\Psi_n)$ in ${\mathcal{D}}(\Omega)^3$ such that
$$
\Psi_n \to \mathbf{v} \quad \quad\text{in }\;\; L^2(\Omega)^3.
$$
By continuity of $P_0$,
$$
P_0(\Psi_n) \to P_0(\mathbf{v} ) =\mathbf{v} \quad \text{in }
H_0(\operatorname{div}0,\Omega),
$$
with $P_0(\Psi_n) \in P_0({\mathcal{D}}(\Omega)^3 )$. 
This concludes the proof of (ii) and Lemma \ref{lema2.1}. 
\end{proof}

\section{Well-posedness of the problem}

We rewrite system \eqref{eq1.1}-\eqref{eq1.3} in the form
\begin{equation}\label{eq3.1}
\frac{d \Phi (t)}{dt} = A \Phi(t),
\end{equation}
where $\Phi=(\mathbf{E},\mathbf{H},\theta)$ and $A$ is the linear operator
$$
A(\mathbf{E},\mathbf{H},\theta)=\left( -\sigma \epsilon^{-1}\mathbf{E} +  \epsilon^{-1} \nabla
\times \mathbf{H} - \gamma \epsilon^{-1} \nabla \theta,
- \mu^{-1} \nabla \times \mathbf{E}, \operatorname{div}(-\lambda \mathbf{E} + \nabla \theta)
  \right).
$$
Let us consider the Hilbert space 
$W =L^2(\Omega)^3 \times  H_0(\operatorname{div}0,\Omega) \times L^2(\Omega)$
 with the inner product given by
$$
\langle u,v \rangle_W =  \epsilon \lambda (\mathbf{u}_1,\mathbf{v}_1) 
+ \mu \lambda (\mathbf{u}_2,\mathbf{v}_2) + \gamma (u_3,v_3),
$$
and induced norm
$$
\| u\|_{W}^{2}=\epsilon \lambda \| \mathbf{u}_1\|^{2}
+\mu \lambda \| \mathbf{u}_2\|^{2} + \gamma \| u_3\|^{2},
$$
for any $u=(\mathbf{u}_1,\mathbf{u}_2,u_3)$ and
 $v=(\mathbf{v}_1,\mathbf{v}_2,v_3) \in W$.

The domain $D(A)$ of $A$ is the set 
$D(A)=\{ (\mathbf{E},\mathbf{H},\theta) \in  H_0(\operatorname{curl},\Omega)
\times (H(\operatorname{curl},\Omega) 
\cap H_0(\operatorname{div}0,\Omega)) \times H^1_0(\Omega); 
-\lambda \mathbf{E} + \nabla \theta \in H(\operatorname{div},\Omega) \}$,
 where $H^1_0(\Omega)$ denotes the usual Sobolev space.

\begin{remark} \rm
It is easy to see that
\begin{equation}\label{eq3.2}
{\mathcal{D}}(\Omega)^3 \times P_0({\mathcal{D}}(\Omega)^3) 
\times {\mathcal{D}}(\Omega) \subset {D}(A) \subset L^2(\Omega)^3 
\times H_0(\operatorname{div}0,\Omega) \times L^2(\Omega) = W,
\end{equation}
where $P_0$ is the orthogonal projection defined in Lemma \ref{lema2.1}.
\end{remark}

Now we  prove that $A$ is the infinitesimal generator of a $C_0$-semigroup 
of contractions on $W$. The density of $D(A)$ in $W$ follows by \eqref{eq3.2} 
and item (ii) of Lemma \ref{lema2.1}.

\begin{lemma} \label{lema3.1}
$A$ is a dissipative operator on $W$.
\end{lemma}

\begin{proof} 
Let $U=(\mathbf{E},\mathbf{H},\theta) \in D(A)$. So by Gauss and Green's identities
it follows that
\begin{equation} \label{eq3.3}
\begin{aligned}
\langle AU,U \rangle_W
&= \lambda  (-\sigma(x) \mathbf{E} + \nabla \times \mathbf{H} 
- \gamma \nabla \theta , \mathbf{E}) +
\lambda (-\nabla \times \mathbf{E}, \mathbf{H})\\
&\quad + \gamma( \operatorname{div}( \nabla \theta - \lambda \mathbf{E}), \theta)\\
&= -\lambda \int_{\Omega} \sigma(x) |\mathbf{E}|^2\,dx
- \gamma \int_{\Omega} |\nabla \theta|^2\,dx \le 0.
\end{aligned}
\end{equation}
\end{proof}

\begin{lemma} \label{lema3.2}
The range $R(I-A)$ of the operator $I-A$ is $W$.
\end{lemma}

\begin{proof} 
Let $w=(\mathbf{f},\mathbf{g},h) \in W$ and we have to prove that there exists
$ U=(\mathbf{E},\mathbf{H},\theta)$ in $D(A)$ such that  $(I-A)U=w$; that is,
\begin{equation}\label{eq3.4}
\begin{gathered}
\mathbf{E} +\sigma(x) \epsilon^{-1} \mathbf{E} -  \epsilon^{-1} \nabla \times \mathbf{H}
+ \gamma  \epsilon^{-1} \nabla \theta=\mathbf{f}\\
\mathbf{H} + \mu^{-1} \nabla \times \mathbf{E}   =\mathbf{g}\\
\theta  - \operatorname{div}( -\lambda \mathbf{E} + \nabla \theta )= h.
\end{gathered} 
\end{equation}
Replacing the second line in the first line of system \eqref{eq3.4}
 we obtain the equivalent system
\begin{equation}\label{eq3.5}
\begin{gathered}
(1 +\sigma(x) \epsilon^{-1})\mathbf{E} +  \epsilon^{-1} \mu^{-1}\nabla
\times(\nabla \times \mathbf{E})  + \gamma \epsilon^{-1} \nabla \theta
=\mathbf{f} + \epsilon^{-1} \nabla \times \mathbf{g}\\
\theta  - \operatorname{div}( -\lambda \mathbf{E} + \nabla \theta )= h.
\end{gathered}  
\end{equation}
To solve \eqref{eq3.5} we consider the bilinear form
$ a:[H_0(\operatorname{curl},\Omega) \times H^1_0(\Omega)]^2  \to  \mathbb{R}$
defined by 
\begin{align*}
  a((\mathbf{E},\theta),(\Phi,\psi))
& =  \lambda(\mathbf{E},\Phi) + \lambda \epsilon^{-1}(\sigma(x)\mathbf{E},\Phi)
+ \lambda \epsilon^{-1} \mu^{-1}(\nabla\times \mathbf{E} ,\nabla\times \Phi ) \\
&\quad +\lambda \gamma \epsilon^{-1}(\nabla \theta, \Phi)
 +\gamma \epsilon^{-1}(\theta,\psi) - \gamma \epsilon^{-1} 
 (\lambda \mathbf{E} - \nabla \theta , \nabla \psi)
\end{align*}
and the linear form
$F:H_0(\operatorname{curl},\Omega) \times H^1_0(\Omega) \to  \mathbb{R}$
defined by 
$$
F(\Phi,\psi)= \lambda(\mathbf{f},\Phi) 
+ \lambda \epsilon^{-1} (\mathbf{g},\nabla \times \Phi) 
+ \gamma \epsilon^{-1} (h,\psi).
$$

The bilinear form $a$ is coercive, because
\begin{align*}
&a((\mathbf{E},\theta),(\mathbf{E},\theta)) \\
& =  \lambda \| \mathbf{E}\|^2 
 + \lambda \epsilon^{-1}\| \sigma^{1/2}(x)  \mathbf{E}\|^2
 + \lambda \epsilon^{-1}\mu^{-1}\| \nabla\times \mathbf{E}\|^2
  +\gamma \epsilon^{-1}\| \theta\|^2_{H_0^1(\Omega)}  \\
&\ge  C  \|(\mathbf{E}, \theta)\|^2_{H_0(\operatorname{curl},\Omega) \times H^1_0(\Omega)}.
\end{align*}
The bilinear form $a$ is also continuous. Indeed, Cauchy-Schwarz's inequality 
implies
\begin{align*}
&|a((\mathbf{E},\theta),(\Phi, \psi))| \\
&\le \lambda(1+ \sigma_1 \epsilon^{-1}) \| \mathbf{E}\| \|\Phi \|
 + \lambda \epsilon^{-1}\mu^{-1}\| \nabla\times \mathbf{E}\|  \| \nabla\times \Phi\|
 +  \lambda \gamma \epsilon^{-1}\| \nabla\theta\|  \| \Phi\|\\
&\quad  +\gamma \epsilon^{-1}\| \theta\|  \| \psi\| 
  +   \gamma \lambda \epsilon^{-1}\| \mathbf{E}\|  \|\nabla \psi\|
 + \gamma \epsilon^{-1}\|\nabla \theta\|  \|\nabla \psi\|   \\
&\le \lambda(1 + \sigma_1\epsilon^{-1}+\epsilon^{-1}\mu^{-1}) 
 ( \| \mathbf{E}\|_{H_0(\operatorname{curl},\Omega)}
 \| \Phi\|_{H_0(\operatorname{curl},\Omega)} ) \\
&\quad + \gamma \lambda \epsilon^{-1}\| \theta\|_{H^1_0(\Omega)} 
 \| \Phi\|_{H_0(\operatorname{curl},\Omega)}
 +\gamma \epsilon^{-1}\| \theta\|_{H^1_0(\Omega)} \| \psi\|_{H^1_0(\Omega)} \\
&\quad + \gamma \lambda\epsilon^{-1}  \| \mathbf{E}\|_{H_0(\operatorname{curl},
 \Omega)} \| \psi\|_{H_0^1(\Omega)} 
 +  \gamma \epsilon^{-1}\| \theta\|_{H^1_0(\Omega)} \| \psi\|_{H^1_0(\Omega)}   \\
&\le C \Big( \| \mathbf{E}\|_{H_0(\operatorname{curl},\Omega)}^2
 +  \| \theta\|_{H^1_0(\Omega)}^2 \Big)^{1/2} 
  \Big( \|\Phi\|_{H_0(\operatorname{curl},\Omega)}^2  +
  \| \psi\|_{H^1_0(\Omega)}^2 \Big)^{1/2} \\
&= C  \| (\mathbf{E},\theta)\|_{H_0(\operatorname{curl},\Omega) \times H^1_0(\Omega)}
 \| (\Phi,\psi)\|_{H_0(\operatorname{curl},\Omega) \times H^1_0(\Omega)}.
\end{align*}
To prove that $F$ is continuous, we observe that
\begin{align*}
  |F(\Phi, \psi)| 
& \le   \lambda\| \mathbf{f}\| \|\Phi \|  + \lambda  \epsilon^{-1}
 \|\mathbf{g}\|  \| \nabla\times \Phi\| 
+   \gamma  \epsilon^{-1} \| h\|  \| \psi\|  \\
&\le \lambda(1+ \epsilon^{-1} )(\| \mathbf{f}\| 
 + \|\mathbf{g}\|) \| \Phi\|_{H_0(\operatorname{curl},\Omega)}  
 +    \gamma  \epsilon^{-1}\| h\|  \| \psi\|_{ H^1_0(\Omega)}  \\
&\le \big[ \lambda(1+ \epsilon^{-1} )(\| \mathbf{f}\|  
 +  \| \mathbf{g}\|) +\gamma \epsilon^{-1} \| h\| \big]  
\big[ \| \Phi\|_{H_0(\operatorname{curl},\Omega)}^2 
 +  \| \psi\|_{H^1_0(\Omega)}^2\big]^{1/2}   \\
&\le C  \| (\Phi,\psi)\|_{H_0(\operatorname{curl},\Omega) \times H^1_0(\Omega)} .
\end{align*}

By Lax-Milgram's Lemma, there exists a unique 
$(\mathbf{E},\theta) \in H_0(\operatorname{curl},\Omega) \times H^1_0(\Omega) $ such that
\begin{equation} \label{eq3.6}
a((\mathbf{E},\theta),(\Phi, \psi)) = F (\Phi, \psi),\quad \forall
(\Phi, \psi) \in H_0(\operatorname{curl},\Omega) \times H^1_0(\Omega).
\end{equation}
Let
\begin{equation} \label{eq3.7}
 \mathbf{H}= \mathbf{g}- \mu^{-1}\nabla \times \mathbf{E}.
\end{equation}
So $\mathbf{H} \in H_0(\operatorname{div}0,\Omega)$, because
$\mathbf{g} \in  H_0(\operatorname{div}0,\Omega)$ and 
$\nabla \times \mathbf{E} \in  H_0(\operatorname{div}0,\Omega)$ (see \cite[page 35]{3}).

First we consider $\Phi \in  \mathcal{D}(\Omega)^3 $ and
$\psi=0$  in \eqref{eq3.6}. We get
$$
(1 + \sigma(x) \epsilon^{-1})\mathbf{E} +  \epsilon^{-1} \mu^{-1}\nabla
\times (\nabla \times \mathbf{E})  + \gamma \epsilon^{-1} \nabla \theta
= \mathbf{f} + \epsilon^{-1}
\nabla \times \mathbf{g} \quad \text{in }  \mathcal{D}'(\Omega)^3;
$$
that is,
\begin{equation} \label{eq3.8}
(1 + \sigma \epsilon^{-1})\mathbf{E} - \epsilon^{-1} \nabla \times \mathbf{H}
+ \gamma \epsilon^{-1}\nabla \theta
= \mathbf{f}  \quad  \text{in } \mathcal{D}'(\Omega)^3.
\end{equation}
This proves $\mathbf{H} \in H(\operatorname{curl},\Omega)$ and, hence,
 $\mathbf{H} \in H(\operatorname{curl},\Omega) \cap H_0(\operatorname{div}0,\Omega)$.
Now, taking  $\Phi =0$ and  $\psi \in  \mathcal{D}(\Omega) $  
in \eqref{eq3.6} we obtain
\begin{equation} \label{eq3.9}
\theta +  \operatorname{div}( \lambda \mathbf{E} -\nabla \theta)  = h \quad
\text{in }  \mathcal{D}'(\Omega)
\end{equation}
and this proves that $( \lambda \mathbf{E} -\nabla \theta) \in H(\operatorname{div},\Omega)$.
By \eqref{eq3.7}-\eqref{eq3.9} we have $(\mathbf{E},\mathbf{H},\theta) \in D(A)$
and solves \eqref{eq3.4}.
\end{proof}

Using the results before we have the following theorem (see \cite{12}).

\begin{theorem} \label{teor3.1}
Let $(\mathbf{E}_0,\mathbf{H}_0,\theta_0)\in D(A)$. Then problem
\eqref{eq1.1}-\eqref{eq1.6} admits a unique solution  
$\left(\mathbf{E},\mathbf{H},\theta\right)$ such that
\begin{gather*}
\mathbf{E} \in C( [0, + \infty), H_0(\operatorname{curl},\Omega))
\cap C^{1}( [0, + \infty), L^2(\Omega)^3),\\
\mathbf{H} \in C( [0, + \infty), H(\operatorname{curl},\Omega)
\cap H_0(\operatorname{div}0,\Omega) )  
\cap C^{1}( [0, + \infty), H_0(\operatorname{div}0,\Omega)),\\
\theta  \in C( [0, + \infty), H_0^1(\Omega) )  \cap 
C^{1}( [0, + \infty), L^2(\Omega)),\\ 
\lambda \mathbf{E} - \nabla \theta \in C( [0, + \infty), H(\operatorname{div},\Omega) ) .
\end{gather*}
\end{theorem}

To obtain the stability of solution of system \eqref{eq1.1}-\eqref{eq1.6} 
we need to a more regular solution. To this, we consider the spaces
$$
\mathbb{H}_1(\Omega)
= H(\operatorname{curl}0,\Omega) \cap H_0(\operatorname{div}0,\Omega),
$$
where $H(\operatorname{curl}0,\Omega)=\{\mathbf{u} \in L^2(\Omega)^3:
 \nabla \times \mathbf{u}=0\}$,
and
$$
\mathcal{V}_H = L^2(\Omega)^3 \times \mathbb{H}_1(\Omega)^{\bot}
\times L^2(\Omega),
$$
where $\mathbb{H}_1(\Omega)^{\bot}$ is the orthogonal complement
of the space $\mathbb{H}_1(\Omega)$ in $L^2(\Omega)^3$.

We have the following existence result on strong solutions of system 
\eqref{eq1.1}-\eqref{eq1.6}.

\begin{theorem} \label{teor3.2}
Let $\left(\mathbf{E}_0,\mathbf{H}_0,\theta_0\right)\in D(A) \cap \mathcal{V}_H$.
Then the solution $\left(\mathbf{E},\mathbf{H},\theta\right)$ of \eqref{eq1.1}-\eqref{eq1.6}
 obtained in Theorem \ref{teor3.1} satisfies 
 $\left(\mathbf{E},\mathbf{H},\theta\right)\in D(A) \cap \mathcal{V}_H$ for all $t>0$.
\end{theorem}

\begin{proof} 
It is sufficient to prove that $\mathbf{H} \in \mathbb{H}_1(\Omega)^{\bot}$.
To this, we consider $\mathbf{h} \in \mathbb{H}_1(\Omega)$.
From \eqref{eq1.2} we obtain
\begin{equation}\label{eq3.10}
\int_{\Omega}\mu \mathbf{H}_t \cdot \mathbf{h}\,dx
+ \int_{\Omega}\nabla\times \mathbf{E} \cdot \mathbf{h}\,dx=0.
\end{equation}
Green's formula gives us
\[
\int_{\Omega}\nabla\times \mathbf{E} \cdot \mathbf{h}\,dx
= \int_{\Omega} \mathbf{E} \cdot( \nabla \times \mathbf{h})\,dx
+ \int_{\Gamma} (\eta \times E)\cdot \mathbf{h} \,d\Gamma = 0.
\]
The above identity and \eqref{eq3.10} give us  
$(\mu \mathbf{H},\mathbf{h}) = (\mu \mathbf{H}_0,\mathbf{h})=0 $.
So $\mathbf{H} \in \mathbb{H}_1(\Omega)^{\bot}$.
\end{proof}

\section{Orthogonal decomposition}

Using the standard ``Hodge'' orthogonal decomposition of $L^2(\Omega)^3$ 
(see \cite{1,3,13}) we can write
\begin{equation}\label{eq4.1}
\mu \mathbf{H}= \nabla q +\mathbf{h}_1 + \nabla \times \Psi,
\end{equation}
where $q \in H^1(\Omega)$, $\mathbf{h}_1  \in \mathbb{H}_1(\Omega)$,
$\Psi \in H^1(\Omega)^3 \cap H_0(\operatorname{curl},\Omega) 
\cap H(\operatorname{div}0,\Omega)$ and  
$ \int_{\Gamma}\Psi \cdot \eta \,d\Gamma =0$.

Since $\mathbf{H} \in \mathbb{H}_1(\Omega)^{\bot} \cap
 H_0(\operatorname{div}0,\Omega)$, we have $\mathbf{h}_1=0$ and $\nabla q=0$ 
(see \cite{1}), so
\begin{equation}\label{eq4.2}
\mu \mathbf{H}= \nabla \times \Psi.
\end{equation}

\begin{remark} \rm
It is well know (see \cite{1,3}) that for all 
$\mathbf{v} \in  H(\operatorname{div}0,\Omega) \cap H_0(\operatorname{curl},\Omega) $ 
is valid the  inequality
$$
\|\mathbf{v}\| \le C \|\nabla \times \mathbf{v}\|,
$$
where $C$ is a real positive constant. In our case, we obtain
\begin{equation} \label{eq4.3}
\|\Psi\| \le C \|\nabla \times \Psi\| = C\|\mu \mathbf{H}\|.
\end{equation}
\end{remark}

Now, we will study  the $L^2(\Omega)^3$ decomposition of the electric 
field $\mathbf{E}$. In fact, we have (see \cite{1})
\begin{equation} \label{eq4.4}
\mathbf{E}= - \nabla p + \mathbf{B},
\end{equation}
where $p \in H_0^1(\Omega)$ and $\mathbf{B} \in H(\operatorname{div}0,\Omega)$.

From  equation \eqref{eq1.2} and decomposition \eqref{eq4.2} of $\mathbf{H}$ we obtain
\begin{equation}\label{eq4.5}
0= \mu \mathbf{H}_t + \nabla \times \mathbf{E}
= \nabla \times \Psi_t + \nabla \times \mathbf{E}
= \nabla \times(\Psi_t + \nabla p + \mathbf{E}).
\end{equation}
Also,
\begin{equation}\label{eq4.6}
\operatorname{div}(\Psi_t + \nabla p + \mathbf{E})
= \operatorname{div}(\Psi_t) + \operatorname{div}(\mathbf{B})=0,
\end{equation}
because $\Psi,\mathbf{B} \in H(\operatorname{div}0,\Omega)$.

The  last two equalities give us
$$
\Psi_t + \nabla p + \mathbf{E} \in H(\operatorname{curl}0,\Omega) \cap  H(\operatorname{div}0,\Omega).
$$
Now, we observe that $\Psi_t \in H_0(\operatorname{curl},\Omega)$,
$\mathbf{E} \in  H_0(\operatorname{curl},\Omega)$ and,
since $p \in H_0^1(\Omega)$, 
$\nabla p \in H_0(\operatorname{curl}0,\Omega):=H(\operatorname{curl}0,\Omega) 
\cap H_0(\operatorname{curl},\Omega)$ (see \cite{3}). So
\begin{equation}\label{eq4.7}
\Psi_t + \nabla p + \mathbf{E} \in  \mathbb{H}_2(\Omega),
\end{equation}
where $\mathbb{H}_2(\Omega)
=  H_0(\operatorname{curl}0,\Omega) \cap H(\operatorname{div}0,\Omega)$.
From \eqref{eq4.7} we can write
\begin{equation}\label{eq4.8}
 \mathbf{E}= -\nabla p - \Psi_t + \mathbf{h}_2,
\end{equation}
where  $\mathbf{h}_2 \in \mathbb{H}_2(\Omega)$.

Finally,  we can see that
\begin{equation}\label{eq4.9}
\|\mathbf{E}\|^2 = \|\nabla p\|^2 + \|\Psi_t\|^2+ \|\mathbf{h}_2\|^2,
\end{equation}
because $\nabla p$, $\Psi_t$ and $\mathbf{h}_2$ are two by two 
orthogonal vectors in $L^2(\Omega)^3$ (see \cite{13}).

\section{Exponential decay}

In this section we obtain the exponential decay of the solution 
of system \eqref{eq1.1}-\eqref{eq1.6} obtained in section 3. 
To this, we use a suitable Lyapunov functional and suppose that 
$\sigma$ satisfies hypothesis \eqref{eq1.7}. First we present some 
technical lemmas and at the end of the section we prove the main result 
of this paper.

\begin{lemma} \label{lema5.1}
Suppose $\left(\mathbf{E}_0,\mathbf{H}_0,\theta_0\right)\in D(A) \cap \mathcal{V}_H$ and let
$\left(\mathbf{E},\mathbf{H},\theta\right)$ solution of system \eqref{eq1.1}-\eqref{eq1.6}
obtained in Theorem \ref{teor3.2}. Let 
$$
\mathcal{E} (t)\equiv \frac{1}{2} \int_{\Omega}(\lambda \epsilon |\mathbf{E}|^2
 + \lambda \mu |\mathbf{H}|^2 + \gamma|\theta|^2)\,dx.
$$ 
Then
 $$
\frac{d {\mathcal{E} (t)}}{dt} =  - \lambda  \int_{\Omega}\sigma(x)|\mathbf{E}|^2 \,dx
- \gamma  \int_{\Omega}|\nabla \theta|^2 \,dx\leq 0.
$$
\end{lemma}

The proof of the above lemma follows directly from the system 
\eqref{eq1.1}-\eqref{eq1.6} using straightforward calculation.

\begin{lemma} \label{lema5.2}
Let $G(t)=\mathcal{E} (t) - \delta F(t)$, where $\mathcal{E} (t) $ is 
defined in Lemma \ref{lema5.1},
\[
F(t) =    \epsilon \int_{\Omega}\mathbf{E} \cdot \Psi\,dx ,
\]
 where $\mu \mathbf{H}= \nabla \times \Psi$,  and $\delta$ is a positive parameter
to be specified later. We have
\begin{itemize}
\item[(i)]  
\[
\frac{d F(t)}{dt} = \mu \int_{\Omega}|\mathbf{H}|^2 \,dx
 -\epsilon \int_{\Omega}|\Psi_t|^2 \,dx 
- \int_{\Omega}\sigma(x)\mathbf{E} \cdot \Psi \,dx ;
\]

\item[(ii)] $\frac{1}{2} \mathcal{E} (t) \le G(t) \le 2 \mathcal{E} (t)$.
\end{itemize}
\end{lemma}

\begin{proof}
 To prove (i), from \eqref{eq1.1} and \eqref{eq4.8}, we observe that 
\begin{align*}
  \frac{d F(t)}{dt}  
& =   \epsilon\int_{\Omega}\mathbf{E} \cdot \Psi_t \,dx
+  \epsilon\int_{\Omega}\mathbf{E}_t \cdot \Psi \,dx  \\
&=  \epsilon\int_{\Omega}(-\nabla p - \Psi_t + \mathbf{h}_2) \cdot \Psi_t \,dx 
 +  \int_{\Omega}(\nabla \times \mathbf{H} - \sigma(x) \mathbf{E}
 - \gamma \nabla \theta) \cdot \Psi \,dx \\
&= -\epsilon\int_{\Omega}|\Psi_t|^2 \,dx  
 +  \int_{\Omega} \nabla \times \mathbf{H}  \cdot \Psi \,dx
 - \int_{\Omega} \sigma(x)\mathbf{E} \cdot \Psi \,dx -
\gamma  \int_{\Omega} \nabla \theta \cdot \Psi \,dx  \\
&= -\epsilon\int_{\Omega}|\Psi_t|^2 \,dx  
 +  \int_{\Omega} \mathbf{H}  \cdot \nabla \times \Psi \,dx
 - \int_{\Omega} \sigma(x)\mathbf{E} \cdot \Psi \,dx +
\gamma  \int_{\Omega} \theta\, div \,\Psi \,dx  \\
&=  -\epsilon\int_{\Omega}|\Psi_t|^2 \,dx  
 + \mu\int_{\Omega}|\mathbf{H}|^2 \,dx - \int_{\Omega} \sigma(x)\mathbf{E} \cdot \Psi \,dx,
\end{align*}
because $\Psi\in H_0(\operatorname{curl},\Omega) \cap H(\operatorname{div}0,\Omega)$,
 $p\in H_0^1(\Omega)$ and $\mathbf{h}_2 \in \mathbb{H}_2(\Omega)$.

To prove (ii), we use the Cauchy-Schwarz's inequality and \eqref{eq4.3}:
\begin{align*}
|G(t) - \mathcal{E} (t)|  
& =  \delta |F(t) | \le \delta \epsilon  \|\mathbf{E}\| \|\Psi\| \\
&\le  \frac{\delta \epsilon }{2}\left( \|\mathbf{E}\|^2  + \|\Psi\|^2 \right)  \\
&\le  \frac{\delta \epsilon }{2}\left(  \|\mathbf{E}\|^2  + C^2\|\mu \mathbf{H}\|^2 \right)  \\
&= \frac{\delta}{2 \lambda}\Big( \int_{\Omega}\lambda\epsilon |\mathbf{E}|^2\,dx
 + C^2 \mu \epsilon \int_{\Omega}\lambda\mu |\mathbf{H}|^2 \,dx\Big)   \\
&\le \delta C_1 \mathcal{E} (t),
\end{align*}
where
$$
C_1= \max \Big\{ \frac{1}{ \lambda},\frac{C^2\mu\epsilon }{\lambda}\Big\}.
$$
The conclusion follows by choosing $\delta$ sufficiently small such that
\begin{equation}\label{eq5.1}
\delta C_1\le\frac{1}{2}.
\end{equation}
\end{proof}

Now, we prove the main result of this paper.

\begin{theorem} \label{teor2}
Suppose $\left(\mathbf{E}_0,\mathbf{H}_0,\theta_0\right)\in D(A) \cap \mathcal{V}_H$ and $\sigma$
 satisfies \eqref{eq1.7}. Then the total energy $\mathcal{E}(t)$ of  problem 
\eqref{eq1.1}-\eqref{eq1.6}, defined in Lemma \ref{lema5.1}, satisfies
$$
\mathcal{E}(t) \le \beta \mathcal{E}(0)\exp(-\alpha t),
$$
where $\beta$ and $\alpha$ are positive constants.
\end{theorem}

\begin{proof} 
From Lemmas \ref{lema5.1} and \ref{lema5.2} we have
\begin{align*}
 \frac{d G(t)}{dt}  
& =  -\lambda \int_{\Omega}\sigma(x) |\mathbf{E}|^2\,dx
 -  \gamma  \int_{\Omega} |\nabla \theta|^2\,dx  \\
&\quad - \delta \int_{\Omega}\mu |\mathbf{H}|^2\,dx
 + \delta \int_{\Omega}\sigma(x) \mathbf{E}\cdot \Psi\,dx
 +\delta \epsilon\int_{\Omega}|\Psi_t|^2\,dx
\end{align*}
and from \eqref{eq1.7}, \eqref{eq4.3}, \eqref{eq4.9} and Poincar\'e Inequality,
\begin{align}
 \frac{d G(t)}{dt}  
& \le   -\frac{\sigma_0}{\epsilon} \int_{\Omega}\lambda\epsilon|\mathbf{E}|^2\,dx
-  C_0  \int_{\Omega} \gamma| \theta|^2\,dx 
-  \frac{\delta}{\lambda} \int_{\Omega}\lambda\mu|\mathbf{H}|^2\,dx  \\
&\quad + \frac{\delta}{2} \left(\frac{\sigma_1^2}{\kappa}\|\mathbf{E}\|^{2}
 +C^2\kappa \|\mu \mathbf{H}\|^{2}\right) +\delta \epsilon\int_{\Omega}|\mathbf{E}|^2\,dx \\
&=-\Big[ \frac{\sigma_0}{\epsilon}
 -\Big(\frac{\sigma_1^2}{2\kappa\epsilon\lambda}+\frac{1}{\lambda}\Big)
 \delta\Big]\int_{\Omega}\lambda\epsilon|\mathbf{E}|^2\,dx
 -C_0  \int_{\Omega} \gamma| \theta|^2\,dx \\
&- \delta\Big( \frac{1}{\lambda}-\frac{1}{2\lambda}C^2\mu \kappa \Big)
 \int_{\Omega}\lambda\mu |\mathbf{H}|^2\,dx.
\end{align}
We choose $\kappa>0$ such that
$$
C_2\equiv \frac{1}{\lambda}-\frac{1}{2\lambda}C^2\mu \kappa>0
$$
and $\delta>0$ small satisfying \eqref{eq5.1} and
$$
C_3\equiv \frac{\sigma_0}{\epsilon}
-\big(\frac{\sigma_1^2}{2\kappa\epsilon\lambda}+\frac{1}{\lambda}\big)\delta>0.
$$
Thus
\begin{equation}
  \frac{d G(t)}{dt}
 \le   -C_3\int_{\Omega}\lambda\epsilon|\mathbf{E}|^2\,dx
 - \delta C_2\int_{\Omega}\lambda\mu |\mathbf{H}|^2\,dx
 -C_0  \int_{\Omega} \gamma| \theta|^2\,dx
 \le-C_4\mathcal{E}(t),
\end{equation}
where
$C_4=\min\{2C_3,2\delta C_2,2C_0 \}$.
From Lemma \ref{lema5.2} and the above inequality we obtain
\[
\frac{d G(t)}{dt} \le - \frac{C_4}{2}G(t) \quad\text{and}\quad 
G(t)  \le G(0)\exp(-\frac{C_4}{2}t).
\]
Finally, we conclude that
$$
\mathcal{E}(t) \le 2G(t) \le  4 \mathcal{E}(0)\exp(-\frac{C_4}{2}t).
$$
\end{proof}

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\end{document}